two like charged, infinitely long parallel wires with the same linear charge density of 3*10^-8 C/m are 2 cm apart. the work done against electrostatic force per unit length to be done in bringing them closer by 1 cm is X/100 J/m: find the value of X.

Dear Student,

Please find below the solution to the asked query:

The electric field at distance x from an infinite line of charge (linear charge density λ) is

$E=\frac{1}{4\pi {\epsilon }_0}\frac{2\lambda }{x}$
So, force per unit length is,

$F=\frac{1}{4\pi {\epsilon }_0}\frac{2{\lambda }^2}{x}$

Therefore, the work done is,

$$\begin{gathered} W=-\int F.dx={\int }_{1 cm}^{2 cm}\frac{1}{4\pi {\epsilon }_0}\frac{2{\lambda }^2}{x}dx=\frac{2{\lambda }^2}{4\pi {\epsilon }_0}{\left[-\frac{1}{x^2}\right]}_{1 cm}^{2 cm} \\ \Rightarrow W=9\times {10}^9\times 2\times 9\times {10}^{-16}\times \left[-\frac{1}{4\times {10}^{-4}}+\frac{1}{{10}^{-4}}\right] \\ \Rightarrow W=162\times {10}^{-7}\times {10}^4\times \frac{3}{4} \\ \Rightarrow W=\frac{243}{2}\times {10}^{-3}\Rightarrow W=\frac{243}{2000} \Rightarrow W=\frac{24.3}{200} \\ \Rightarrow W=\frac{12.15}{100} \raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{$C$}\right. \\ \therefore x=12.15 \end{gathered}$$

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